Centroid and First Moment of Area

The centroid of an area is going to be located at the same position regardless of the area's position with respect to the reference coordinate system.

If an area has an axis of symmetry, the first moment of area with respect to that axis is zero and the centroid is located along that axis.


Composite Area

1.
In order to to find the centroid of a composite area, the composite area must be subdivided into basic shapes. The basic shapes must have the same reference coordinate system as the composite area.

Example


2.
When the basic shapes are obtained, the following must be found for each one:
  • area (A)
  • horizontal centroid distance ()
  • vertical centroid distance ()
3.
Take the product of area and horizontal centroid distance (A) for each basic shape.
Similarly, take the product of area and vertical centroid distance (A) for each basic shape.

4.
Sum the areas for all the basic shapes (A). Sum the products of area and horizontal centroid distance for all the basic shapes (A), which is the same thing as the first moment (Qy) of the composite area with respect to the y-axis. Sum the products of area and vertical centroid distance for all the basic shapes (A), which is the same thing as the first moment (Qx) of the composite area with respect to the x-axis.

5.
The centroid is then found by calculating and , where:
= (A)/(A)
= (A)/(A)